Let be a free monoid with involution on . A word is said to be cyclically reduced if every cyclic conjugate of it is reduced. In other words, is cyclically reduced iff is a reduced word and that if for some words and , then .
For example, if , then words such as
are cyclically reduced, where as words
are not, the former is reduced, but of the form , while the later is not even a reduced word.
Remarks. The concept of cyclically reduced words carries over to words in groups. We consider words in a group .
If a word is cyclically reduced, so is its inverse and all of its cyclic conjugates.
A word is a cyclic reduction of a word if for some word , and is cyclically reduced. Clearly, every word and its cyclic reduction are conjugates of each other. Furthermore, any word has a unique cyclic reduction.
Every group has a presentation such that
is cyclically reduced (meaning every element of is cyclically reduced),
closed under inverses (meaning if , then ), and
closed under cyclic conjugation (meaning any cyclic conjugate of an element in is in ).
Furthermore, if is finitely presented, above can be chosen to be finite.
Every group has a presentation . There is an isomorphism from to , where is the free group freely generated by , and is the normalizer of the subset in . Let be the set of all cyclic reductions of words in . Then , since any word not cyclically reduced in is conjugate to its cyclic reduction in , and hence in . Next, for each , toss in its inverse and all of its cyclic conjugates. The resulting set is still cyclically reduced. Furthermore, satisfies the remaining conditions above. Finally, , as any cyclic conjugate of a word is clearly a conjugate of . Therefore, has presentation .
If is finitely presented, then and above can be chosen to be finite sets. Therefore, and are both finite. is finite because the number of cyclic conjugates of a word is at most the length of the word, and hence finite. ∎
|Date of creation||2013-03-22 17:34:04|
|Last modified on||2013-03-22 17:34:04|
|Last modified by||CWoo (3771)|